The door - or sometimes the gate - is one of the great images that John quotes in his Gospel to help us understand what Jesus is like. The specific door (or gate) to which Jesus refers is one on a sheepfold and so it is to do with taking care of the sheep. The door is vitally important because we need to go through it in order to get where we need to be. Sometimes we need to be outside, and at other times inside - but going through the door is needed to get us to the other side.
Stephen Verney (in "Water into Wine", Fount, 1985) recognises the important role of the door - "A door has two sides, an inside and an outside. In the figure of speech Jesus has used, one side of the door is the courtyard of the sheepfold and the other side is the open country. In our own houses, one side of the front door is home and the other side is the street. The truth of I AM is also a door with two sides - one side is a man on earth and the other side is God in heaven, and through that door of I AM the love of the human race goes up to God, and the Love of God comes down on the human race" (p. 104).
Similarly Lesslie Newbigin says (in "The Light Has Come" Eerdmans, 1982), "The door is a universally evocative symbol. It is the way of access from one world to another and therefore also the way by which the reality of that world may be communicated to this" (p. 127).
The idea of the door thus takes us into the realms of communication and relationship. If we are to be God's people, we need to be that.